Unknown

206 7. Approximations and Inequalities

The coefficients a, b are to be determined from the conditions that

~(50) = 1.32894 and r(60) = 1.32718. Verify that this leads to a =

-0.000176 and b = 1.33774. What value of r(54) will this yield?

3. Let f(z) be a real-valued function defined on the closed interval
   [a,b] = {x : a 5 2: 5 b}. Show that f(z) is linear (i.e. a polyno-
   mial of degree 1) if and only if

f(x) = (b - u)-~[(z - u>f(b> + (b - 4f(41.

Verify that, if 2 = (1 - t)u + tb, this condition can be rewritten

f(x) = Cl- W(u) + tf(b).

4. (a) On a graph, plot the points (T, r(T)) and join the points by a
   smooth curve. Observe that the ,relationship between T and r(T)
   is probably not linear and argue that the linear approximation
   has probably underestimated the true value of r(54).
   (b) In an attempt to improve our estimate of r(54), we can try a
   quadratic formula r(T) = uT2 + bT + c, for values of T near 54.
   Since there are three coefficients to be found, we use information
   about r for three values of T. We take those values closest to 54,
   namely 40, 50 and 60. Show that this leads to the formula

r(T) = -0.00000095T2 - 0.0000715T + 1.33489.

What value of r(54) does this yield?

(c) The experimental value of ~(54) is 1.32827. Is this what you

would expect?

5. In the general interpolation situation, we have a function f whose
   values are known at n + 1 points:

f(ai)=bi (i=O,l,2 ,..., n).

To estimate its values elsewhere, we could take in its place a polyno-

mial which agrees with the function f at the ei. As in the example of

the index of refraction, we try to make the degree of the polynomial

as small as possible.

In this connection, we have the result that there is a unique polyno-

mial p(t) such that degp(t) 5 n and p(si) = bi for i = 0, 1,2,... ,n.

(cf. Exercise 4.6.8).

There is a straightforward way of constructing this polynomial. First,

we need some building blocks. With the help of the Factor Theorem,

determine, for each i = 0, 1,... , n, that there is a polynomial pi(t) of

degree not exceeding n for which

pi(q) = 1 pi(q) = 0 (i # j).